Let's follow the steps to construct triangle LMN, add the necessary points, and use the slope formula to confirm that line segment QR is parallel to LM.
Part A: Create Triangle LMN
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Choose Coordinates for Points L, M, and N:
- Let point L be at (0, 0).
- Let point N be at (0, 3) (since LN is a vertical line segment).
- Let point M be at (4, 3) (to create a horizontal line segment MN).
Now we have:
- Point L (0, 0)
- Point N (0, 3)
- Point M (4, 3)
Part B: Create QR parallel to LM
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Create QR:
- Since Q needs to be on line LN, we can choose point Q at (0, 1) (1 unit above L on vertical line LN).
- Since R needs to be on line MN, we can choose point R at (4, 1) (1 unit below M on horizontal line MN).
Now we have:
- Point Q (0, 1)
- Point R (4, 1)
Part C: Label Points with Coordinates
- L (0, 0)
- M (4, 3)
- N (0, 3)
- Q (0, 1)
- R (4, 1)
Part D: Use the Slope Formula to Confirm QR is Parallel to LM
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Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
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Find Slope of LM:
- Points L (0, 0) and M (4, 3): \[ m_{LM} = \frac{3 - 0}{4 - 0} = \frac{3}{4} \]
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Find Slope of QR:
- Points Q (0, 1) and R (4, 1): \[ m_{QR} = \frac{1 - 1}{4 - 0} = \frac{0}{4} = 0 \]
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Conclusion
- The slope of line segment LM is \( \frac{3}{4} \).
- The slope of line segment QR is \( 0 \), meaning it is horizontal.
Since the slopes of LM and QR are different (and QR is horizontal while LM is not), QR is not parallel to LM. Instead, the segments are perpendicular (with LM having a positive slope and QR being horizontal).
If you intended for QR to be parallel to LM, we should consider having R and Q at different y-coordinates while keeping the x-coordinates the same, leading to both having the same slope, \( \frac{3}{4} \).
Would you like to correct the points for QR accordingly?